{"id":"8ccc16de-46f4-4829-8c21-90509ef07bf7","arxiv_id":"2506.16057","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Kawaguchi-Silverman conjecture is proved for projective bundles over abelian varieties or smooth projective varieties of Picard number one, via new structure theorems for endomorphisms of Fano contractions.","lead":"A team of mathematicians proves a long-standing growth-rate conjecture for certain bundles of projective spaces over abelian varieties or one-dimensional-base-like varieties. The proof uses the geometry of the ramification divisor to build a smaller dynamical system where the conjecture was already known.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 6.3 rests on Claim 6.4, whose equivariant cover splitting D into exactly s components is asserted only by reference to the unpublished preprint [MZ24, Lemma 5.1]; the paper's own §1.4 flags the singularities of these covers as the main obstacle.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing point: Claim 6.4, specifically the existence of the equivariant cover with the required splitting, is deferred to the unpublished reference [MZ24, Lemma 5.1]. My reading of the manuscript confirms this. Section 6 is the heart of the paper: Theorem 6.3's proof reduces positivity of κ_f(X, R_f) to Claim 6.4, and Theorems 1.3 and 1.5 then follow from Theorem 6.3 plus the cone decomposition and dynamical Iitaka fibration machinery. Everything else in the paper is comparatively solid: Theorem 1.6 has a real proof using logarithmic forms and semistability; the cone decomposition theorem has a proof; the deductions in Sections 7 and 8 are straightforward given the positivity theorem and the cited KSC results. I also agree with the reader's secondary observation about the sign typo in Lemma 7.1; it is likely a typo because with the displayed sign a(δ_f − 1) < 0, contradicting pseudo-effectiveness of R_f, whereas a(1 − δ_f)D is positive and consistent with the cone decomposition. Thus the single most load-bearing concern is the unproved external dependence in Claim 6.4. It is not an internal inconsistency, and the claim is plausible — the reference is by the same school, and the strategy is believable — but as written, the central theorem's correctness is conditional on a preprint the authors have not incorporated. The paper itself states in §1.4 that the singularities of the equivariant covers are the main obstacle to generalizing the structure theorems, which is a limitation admission that strengthens the concern. Because the reader already issued a CONDITIONAL verdict and the concern does not by itself demonstrate falsehood, I recommend keeping the verdict unchanged. A reasonable reviewer could also argue for UNVERDICTED until [MZ24] is available and checked, but conditional acceptance with a requirement to supply the proof of Claim 6.4 is the most calibrated outcome.","tokens_in":18482,"tokens_out":12522,"duration_ms":152289,"concrete_test":"Read [MZ24, Lemma 5.1] (arXiv:2408.00566v1) and check whether it constructs a cover pY → Y with the following properties: (i) a lift pg : pY → pY making the system equivariant; (ii) the pullback p_X^{-1}(D) is reduced with exactly s = dim X_y + ρ(X_y) irreducible components; (iii) after the equivariant resolution used in the ρ(Y)=1 case, the number of components remains exactly s. If the lemma only splits a divisor on Y, or if it does not control the number of components of p_X^{-1}(D), then Claim 6.4 is not proved and Theorem 6.3 lacks a key step. A complementary check is to work out a concrete example, e.g. Y = P^1, X = F_e with D a horizontal section and pY → Y a double cover: verify that p_X^{-1}(D) has exactly s components and that pπ^{-1}(pQ) is prime for each prime divisor pQ on pY.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of the positivity theorem 6.3 — and therefore of Theorems 1.3 and 1.5 and Corollaries 1.8 and 1.9 — depends on Claim 6.4. That claim asserts an equivariant finite (or generically finite) cover pY → Y such that p_X^{-1}(D) splits into exactly s irreducible components and pπ^{-1}(pQ) is prime for every prime divisor pQ on pY. The proof of Claim 6.4 is one sentence: 'By the same strategy as in the proof of [MZ24, Lemma 5.1], there exists ...' No construction, no verification of the exact number of components, and no proof of equivariance of the lifted endomorphism pg is given. [MZ24] is an unpublished preprint (arXiv:2408.00566v1), and the paper explicitly admits in §1.4: 'Currently, the main obstacle to generalizing the two structure theorems is the singularities of equivariant covers appeared in the proof of Claim 6.4.' This is an in-scope limitation statement: the authors themselves identify the step as not settled. If [MZ24, Lemma 5.1] does not produce a cover satisfying all four conditions — in particular, if it only splits a divisor on Y rather than the horizontal divisor D on X, or if the number of components is not controlled — then the contradiction argument in Theorem 6.3 collapses: the rank inequality from Lemma 6.2 cannot be combined with the s components, and the existence of a nontrivial linear relation pD ≡ 0 (hence κ > 0) no longer follows. The verification of condition (4) given in the two cases of Claim 6.4 is plausible: in the abelian case it uses Lemma 6.1 and codimension of Σ, and in the ρ(Y)=1 case smoothness plus connected fibres is close to a proof. But the existence part remains unproved in this paper. A related but secondary issue is the sign typo in Lemma 7.1, where a(δ_f − 1)D should presumably be a(1 − δ_f)D; this appears typographical rather than conceptual.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops dynamical Iitaka theory for the ramification divisor of surjective endomorphisms on Fano contractions (Mori fibre spaces). The main results (Theorems 1.3 and 1.5) assert that, under the hypotheses δ_f > δ_{f|Y} and suitable conditions on the base Y, the ramification divisor satisfies f^*R_f ≡ δ_f R_f, and the f-Iitaka fibration of R_f induces an f-equivariant dominant rational map to a lower-dimensional variety Z on which f|_Z is δ_f-polarized. These structure theorems are then applied to prove the Kawaguchi-Silverman conjecture for projective bundles over abelian varieties or varieties of Picard number one (Corollaries 1.8 and 1.9). The proof strategy combines cone decompositions (Section 4), a toric characterization of general fibres via logarithmic sheaves (Theorem 1.6, Section 5), and a positivity theorem for κ_f(X,R_f) (Theorem 6.3, Section 6). The paper is clearly organized and the arguments in Sections 4 and 5 are detailed, but the crucial Claim 6.4, needed for Theorem 6.3, is not proved in the manuscript and depends on an unpublished preprint.","tokens_in":18879,"tokens_out":15254,"duration_ms":161342,"significance":"If the structure theorems are correct, they would verify Question 1.1 and Question 1.2 in the Fano-contraction setting and provide a conceptual proof of KSC for new classes of varieties, extending earlier results for projective bundles over elliptic curves and Fano varieties. The paper also contains potentially useful technical contributions: the cone decompositions (Theorem 4.5) and the toric characterization theorem (Theorem 1.6) are interesting in their own right. The writing is generally careful, and the non-disputed parts contain substantial and detailed proofs. However, the central theorems depend on Claim 6.4, whose proof is deferred to an unpublished preprint, and on Theorem 2.15 from the same preprint; as a consequence, the main claims are not yet established in the present manuscript.","major_comments":[{"comment":"Claim 6.4 is load-bearing for Theorem 6.3 and therefore for Theorems 1.3 and 1.5, but its proof is a single sentence referring to [MZ24, Lemma 5.1], an unpublished preprint. The claim requires the existence of an equivariant finite (or generically finite) cover pY → Y such that the pullback of the horizontal ramification divisor D splits into exactly s irreducible components, and such that pπ satisfies the hypotheses of Lemma 6.2. No construction, no verification of the exact component count, and no proof of equivariance of the lifted endomorphism are given in this paper. The manuscript itself states in §1.4 that the singularities of these equivariant covers are the main obstacle to generalizing the structure theorems. As written, the positivity statement κ_f(X,R_f) > 0 is not established, and the main structure theorems do not follow.","section":"Section 6, Claim 6.4"},{"comment":"The sign in the ramification divisor formula appears to be incorrect. From K_X = π^*B + aD with a < 0 and f^*D ≡ δ_f D, one obtains R_f = K_X - f^*K_X ≡ π^*(B - g^*B) + a(1 - δ_f)D, not a(δ_f - 1)D. With the printed sign, the effective divisor R_f would have a negative D-coefficient, contradicting the cone decomposition PE(X) = π^*PE(Y) ⊕ R_{≥0}D from Theorem 4.5. This error affects the proof that B - g^*B is pseudo-effective and must be corrected.","section":"Lemma 7.1"},{"comment":"The proof of Corollary 1.9 in the case δ_f > δ_g > 1 invokes Theorem 2.15, which is the main theorem of the unpublished preprint [MZ24]. Thus the paper's headline application is conditional on another unpublished result. The authors should either prove this case directly or cite a published version of [MZ24].","section":"Corollary 1.9, proof"}],"minor_comments":[{"comment":"The text refers to 'Proposition 6.1' in the proof of Claim 6.4; no Proposition 6.1 is stated in the paper. This should be 'Lemma 6.1'.","section":"Section 6, proof of Claim 6.4 (Theorem 1.3 case)"},{"comment":"The notation f_*D_i appears in the proof of Theorem 6.3; the subsequent equations use the pullback f^*, so f_* should probably be f^* throughout this paragraph.","section":"Section 6, proof of Theorem 6.3"},{"comment":"The displayed isomorphism in Claim 5.2 uses f^* in the text but the surrounding discussion sometimes writes f_*; please clarify the notation.","section":"Section 5, Claim 5.2"},{"comment":"Reference [NZ21] is listed as 'Preprint, 2021' with arXiv number 2310.03313, which appears to be from 2023; please update the year and version information.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript relies heavily on unpublished work of the first author, particularly [MZ24], both for the crucial Claim 6.4 and for Theorem 2.15 used in Corollary 1.9. This is not circular reasoning, but it makes the current version difficult to evaluate. If the authors can supply a full proof of Claim 6.4 (or cite a published version of [MZ24]) and fix the sign error in Lemma 7.1, the paper would be substantially stronger. The scope fit with the journal is good, and the technical content in Sections 4 and 5 is sound and useful."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Open with the punchline: this paper has real content—new structure theorems for endomorphisms on extremal Fano contractions, a clean cone decomposition, and applications to the Kawaguchi–Silverman conjecture. But the main positivity theorem rests on a claim proved only by a one-sentence reference to an unpublished preprint by the first author, and the authors themselves flag that step as the main obstacle. That makes the paper conditional, not settled.\n\nWhat is new and good: Theorem 4.5, the cone decomposition, is self-contained and useful beyond this paper. Theorem 1.6, the toric pair criterion, is a solid relative version of prior work, with a detailed proof using logarithmic forms and semistability. Theorems 1.3 and 1.5 are substantial, and Corollary 1.9 genuinely extends earlier cases from elliptic curves and Fano bases to abelian and Q-abelian varieties. The writing is honest: section 1.4 explicitly states the singularities of equivariant covers are the unresolved obstacle. That is helpful.\n\nSoft spots: the load-bearing point is Claim 6.4 in Section 6. It asserts an equivariant finite cover that splits the horizontal ramification divisor into exactly s components and satisfies the prime-fibre condition. The proof is one sentence: \"By the same strategy as in the proof of [MZ24, Lemma 5.1].\" [MZ24] is an unpublished preprint. The verification of condition (4) in the two cases is sketched, but the existence part is not proved here. If that cover fails, the rank argument in Lemma 6.2 cannot be combined with the s components, and Theorem 6.3 collapses, taking Theorems 1.3 and 1.5 with it. This is a genuine gap in the paper as written, not a minor issue. It is not circular reasoning—the cited lemma is a separate theorem—but the dependence is heavy. A referee will need to check [MZ24, Lemma 5.1] carefully and confirm it produces exactly the cover required. There is also a small sign typo in Lemma 7.1, but that is cosmetic.\n\nOverall: if [MZ24] is correct, the argument likely works. The structure is coherent and the new tools are worth knowing. I would send this to peer review with explicit instructions to scrutinize Claim 6.4 and the reliance on unpublished work. It should not be accepted until that is resolved. For a reading group, it is a good case study of a promising but incomplete proof.","headline":"New structure theorems for endomorphisms on Fano contractions, but the main proof hinges on an unpublished lemma; conditional acceptance.","tokens_in":19533,"tokens_out":3328,"would_cite":true,"duration_ms":32653,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14E30","14M25","20K30","37P55"],"pacs":[],"model":"deepseek-v4-flash","headline":"On Fano contractions, the ramification divisor is shown to be a dynamical eigenvector that forces a polarized quotient, yielding the Kawaguchi-Silverman conjecture for projective bundles over abelian varieties and Picard-number-one bases.","keywords":["dynamical Iitaka dimension","ramification divisor","Fano contraction","Mori fibre space","Kawaguchi-Silverman conjecture","dynamical degree","toric pair"],"falsifier":"Compute $f^*R_f$ for a concrete endomorphism of a $\\mathbb{P}^2$-bundle over an elliptic curve with $\\delta_{f|Y}=1$ and $\\delta_f>1$; a single case where $f^*R_f\\not\\equiv\\delta_f R_f$ would refute Theorems 1.5 and the KSC application. Alternatively, construct an extremal Fano contraction over an abelian variety with $\\delta_f>\\delta_{f|Y}$ admitting no finite equivariant cover that splits the horizontal ramification divisor, which would disprove Claim 6.4 and Theorem 6.3.","tokens_in":18273,"feed_emoji":"📐","tokens_out":12995,"duration_ms":122867,"temperature":0.7,"pith_summary":"The paper aims to show that, on a Mori fibre space (an extremal Fano contraction $\\pi\\colon X\\to Y$), the dynamics of a surjective endomorphism $f$ are controlled by its ramification divisor $R_f$. The two structure theorems state that under the hypotheses of Theorems 1.3 and 1.5, the pullback satisfies $f^*R_f\\equiv\\delta_f R_f$, and the $f$-Iitaka fibration of $R_f$ is an $f$-equivariant dominant rational map $X\\dashrightarrow Z$ with $0<\\dim Z<\\dim X$ whose base is $\\delta_f$-polarized. These constraints are strong enough to prove the Kawaguchi-Silverman conjecture for every smooth projective variety admitting an extremal Fano contraction to an abelian variety, and for every $\\mathbb{P}^n$-bundle over a Q-abelian variety or a smooth projective variety of Picard number one. The reader should care because the results reduce a wide-open arithmetic conjecture to already-established polarized, abelian, and int-amplified cases, and they single out the ramification divisor as the key object when no divisorial contraction exists.","feed_headline":"Dynamical degree conjecture proven for projective bundles","feed_subtitle":"Ramification-divisor structure theorems prove Kawaguchi-Silverman for these Fano contractions.","key_machinery":"The central object is the dynamical Iitaka fibration of the ramification divisor: the $f$-equivariant dominant rational map $\\varphi_{f,R_f}\\colon X\\dashrightarrow Z$ whose base dimension equals the $f$-Iitaka dimension $\\kappa_f(X,R_f)$, built from the span of the iterated pullbacks of $R_f$. The argument is carried by three interlocking mechanisms. First, the cone decompositions $\\mathrm{Nef}(X)=\\pi^*\\mathrm{Nef}(Y)\\oplus\\mathbb{R}_{\\ge0}D$ and $\\mathrm{PE}(X)=\\pi^*\\mathrm{PE}(Y)\\oplus\\mathbb{R}_{\\ge0}D$, where $D$ is the nef, $\\pi$-ample eigenvector of $f^*$ with eigenvalue $\\delta_f$. Second, the toric-pair criterion of Theorem 1.6: if a reduced divisor $D$ satisfies $K_X+D$ being $\\pi$-trivial and $f^*D=qD$ with $q>1$, then the general fibres $(X_y,D|_{X_y})$ are toric pairs. Third, a finite equivariant cover splitting the horizontal part of $R_f$ into exactly the number of components forced by the toric structure, which yields the positivity $\\kappa_f(X,R_f)>0$. Together these force $R_f$ to be non-big, numerically proportional to $\\delta_f R_f$, and to have a nontrivial $f$-Iitaka fibration with $\\delta_f$-polarized base.","core_discovery":"On the paper's own terms: let $f$ be a surjective endomorphism of a smooth projective variety $X$ admitting an extremal Fano contraction $\\pi\\colon X\\to Y$. Under the assumptions of Theorem 1.3 ($Y$ an abelian variety, $f$ with a Zariski dense orbit, $\\delta_f>\\delta_{f|Y}$) or Theorem 1.5 (smooth $f$-equivariant contraction, $\\rho(Y)=1$, $\\delta_{f|Y}=1$), the paper proves that $f^*R_f\\equiv\\delta_f R_f$, where $R_f$ is the ramification divisor and $\\delta_f$ is the first dynamical degree, the spectral radius of $f^*$ on the N\\'eron-Severi space. It then proves that the $f$-Iitaka fibration of $R_f$ gives an $f$-equivariant dominant rational map $\\varphi\\colon X\\dashrightarrow Z$ with $0<\\dim Z<\\dim X$ and $\\delta_f$-polarized $f|_Z$. From these structure theorems it derives that the Kawaguchi-Silverman conjecture holds for any smooth projective variety with an extremal Fano contraction to an abelian variety, and for any $\\mathbb{P}^n$-bundle over a Q-abelian variety (a quasi-\\'etale quotient of an abelian variety) or over a smooth projective variety of Picard number one, by reduction to the known polarized, abelian, and int-amplified cases.","pith_inferences":["The proportionality $f^*R_f\\equiv\\delta_f R_f$ may hold for every equivariant extremal Fano contraction with $\\delta_f>\\delta_{f|Y}$, not only the two cases treated here; testing it on non-abelian or non-smooth bases would show whether the toric-pair and cone-decomposition machinery is the essential mechanism.","Theorem 1.6 is stated for Fano fibrations, but the same hypotheses---a reduced boundary $D$ with $K_X+D$ trivial on fibres and $f^*D=qD$ with $q>1$---appear likely to force toric general fibres in wider fibered settings, a claim that could be checked on explicit equivariant fibrations outside the extremal case.","A self-contained proof of the cover in Claim 6.4, instead of an appeal to [MZ24, Lemma 5.1], would probably extend Theorems 1.3 and 1.5 to singular or non-Q-factorial bases and remove the obstacle the authors identify as the main obstruction to further generalization."],"forward_implications":["The Kawaguchi-Silverman conjecture holds for every smooth projective variety admitting an extremal Fano contraction to an abelian variety (Corollary 1.8).","The Kawaguchi-Silverman conjecture holds for every $\\mathbb{P}^n$-bundle over a Q-abelian variety or a smooth projective variety of Picard number one (Corollary 1.9), generalizing earlier projective-bundle results over elliptic curves.","In the two structure-theorem settings, the paper answers Questions 1.1 and 1.2: the first dynamical degree is preserved along the $f$-Iitaka fibration of $R_f$, and $R_f$ cannot be big without forcing $f$ to be polarized and hence int-amplified, contradicting the degree drop.","When $\\delta_f>\\delta_{f|Y}$, the ramification divisor is $\\delta_f$-eigen and not big, so the dynamics are governed by a lower-dimensional $\\delta_f$-polarized quotient."],"supporting_citations":[{"why":"Supplies the finite equivariant cover that splits the ramification divisor, used in Claim 6.4.","marker":"[MZ24, Lemma 5.1]"},{"why":"Constructs the dynamical Iitaka fibration and its equivariance, on which both structure theorems rely.","marker":"[MZ23b, Theorem 4.6]"},{"why":"The toric characterization that Theorem 1.6 relativizes to Fano fibrations.","marker":"[MZ19, Theorem 1.2]"},{"why":"Gives the KSC reduction through equivariant maps used to deduce Corollaries 1.8 and 1.9.","marker":"[MZ22, Lemma 2.5]"},{"why":"Provides KSC for polarized endomorphisms, the target case for the quotient dynamics.","marker":"[KS16a, Theorem 5]"},{"why":"Provides KSC for int-amplified endomorphisms, used in the Picard-number-one case of Corollary 1.9.","marker":"[MZ24, Main Theorem]"},{"why":"Yields log canonicity and normal-crossing structure of invariant divisors inside the proof of Theorem 1.6.","marker":"[Zha14, Proposition 2.1]"},{"why":"Geometric characterization of toric pairs that concludes Theorem 1.6 from the freeness of the log cotangent sheaf.","marker":"[BMSZ18, Theorem 1.2]"}],"fun_headline_variants":["KSC proven for Fano contractions to abelian varieties","Dynamical Iitaka theory proves KSC on Mori fibre spaces","Ramification divisor structure settles KSC","Structure theorems prove KSC for projective bundles"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the existence of a finite equivariant cover of the base that separates the pieces of the ramification divisor and preserves the property that prime divisors pull back to prime divisors; the paper invokes this cover rather than proving it, and if it does not exist the main structure theorems do not follow.","fun_headline_variants_meta":{"raw":{"variants":["KSC proven for Fano contractions to abelian varieties","Dynamical Iitaka theory proves KSC on Mori fibre spaces","Ramification divisor structure settles KSC","Structure theorems prove KSC for projective bundles"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001104,"raw_usage":{"total_tokens":4574,"prompt_tokens":885,"completion_tokens":3689,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":501,"completion_tokens_details":{"reasoning_tokens":3626}},"tokens_in":501,"tokens_out":3689,"duration_ms":25031,"temperature":1.0,"reasoning_tokens":3626,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T23:47:56.712473+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $f^*R_f$ for a concrete endomorphism of a $\\mathbb{P}^2$-bundle over an elliptic curve with $\\delta_{f|Y}=1$ and $\\delta_f>1$; a single case where $f^*R_f\\not\\equiv\\delta_f R_f$ would refute Theorems 1.5 and the KSC application. Alternatively, construct an extremal Fano contraction over an abelian variety with $\\delta_f>\\delta_{f|Y}$ admitting no finite equivariant cover that splits the horizontal ramification divisor, which would disprove Claim 6.4 and Theorem 6.3.","supporting_citations":[],"review_version":1}